This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingrímsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees. © 2014 Elsevier Inc.
Aval, J. C., Boussicault, A., Bouvel, M., & Silimbani, M. (2014). Combinatorics of non-ambiguous trees. Advances in Applied Mathematics, 56(1), 78–108. https://doi.org/10.1016/j.aam.2013.11.004